Q.If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{c} = \hat{i} - 2\hat{j} + \hat{k}$, find a unit vector parallel to the vector $2\vec{a} - \vec{b} + 3\vec{c}$.

Concept: Unit Vector Verification – first compute the resultant vector, then divide by its magnitude.

Step 1: Compute $2\vec{a} - \vec{b} + 3\vec{c}$

$2\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$

$-\vec{b} = -2\hat{i} + \hat{j} - 3\hat{k}$

$3\vec{c} = 3\hat{i} - 6\hat{j} + 3\hat{k}$

Adding:

$(2-2+3)\hat{i} + (2+1-6)\hat{j} + (2-3+3)\hat{k} = 3\hat{i} - 3\hat{j} + 2\hat{k}$

Step 2: Find magnitude …

Compute the combination term by term:

Adding these:

Its magnitude:

$\left|3\hat i - 3\hat j + 2\hat k\right| = \sqrt{3^2 + (-3)^2 + 2^2} = \sqrt{22}.$ …

Method: Unit Vector Along a Linear Combination of Vectors

Use this when asked for a unit vector parallel to a combination such as $2\vec a-\vec b+3\vec c$.

Steps

Step 1: Evaluate the combination component-wise

Scale each vector by its coefficient, then add like components. Track signs carefully — a subtracted vector flips all of its components.

Step 2: Compute the magnitude

For the result $\vec w=w_x\hat i+w_y\hat j+w_z\hat k$, …

Common Mistakes

Mistake 1: Sign errors on the subtracted vector

Why it's wrong: $-\vec b$ negates all three components of $\vec b$; changing only one is a frequent slip. Correct approach: distribute the minus sign across every component before adding.

Mistake 2: Multiplying only one component by the coefficient …